Nuprl Lemma : two-squares-iff
∀x:ℕ
(∃y,z:ℕ. (((y * y) + (z * z)) = x ∈ ℤ)
⇐⇒ ∃y:ℕisqrt(x) + 1. ((isqrt(x - y * y) * isqrt(x - y * y)) = (x - y * y) ∈ ℤ))
Proof
Definitions occuring in Statement :
isqrt: isqrt(x),
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
prop: ℙ,
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
ge: i ≥ j ,
guard: {T},
sq_type: SQType(T),
less_than': less_than'(a;b),
le: A ≤ B,
uiff: uiff(P;Q)
Lemmas referenced :
isqrt-property,
isqrt_wf,
istype-int,
set_subtype_base,
le_wf,
int_subtype_base,
int_seg_wf,
lelt_wf,
less_than_wf,
nat_wf,
mul_bounds_1a,
decidable__lt,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_formula_prop_eq_lemma,
intformeq_wf,
int_term_value_mul_lemma,
itermMultiply_wf,
mul_preserves_le,
isqrt-of-square,
int_term_value_subtract_lemma,
itermSubtract_wf,
decidable__equal_int,
subtype_base_sq,
false_wf,
int_seg_subtype_nat,
int_seg_properties,
subtract_wf,
equal_wf,
exists_wf,
multiply-is-int-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
Error :inhabitedIsType,
independent_pairFormation,
sqequalRule,
Error :productIsType,
because_Cache,
Error :equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
Error :lambdaEquality_alt,
natural_numberEquality,
independent_isectElimination,
productElimination,
Error :universeIsType,
addEquality,
setElimination,
rename,
multiplyEquality,
Error :equalityIsType1,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation,
unionElimination,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
int_eqEquality,
lambdaEquality,
approximateComputation,
applyLambdaEquality,
cumulativity,
instantiate,
lambdaFormation,
promote_hyp,
pointwiseFunctionality
Latex:
\mforall{}x:\mBbbN{}
(\mexists{}y,z:\mBbbN{}. (((y * y) + (z * z)) = x)
\mLeftarrow{}{}\mRightarrow{} \mexists{}y:\mBbbN{}isqrt(x) + 1. ((isqrt(x - y * y) * isqrt(x - y * y)) = (x - y * y)))
Date html generated:
2019_06_20-PM-02_37_21
Last ObjectModification:
2019_06_12-PM-00_26_06
Theory : num_thy_1
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